9.2 Appendix 2 - Newton's Inverse Square Law

Newton's Inverse Square Law -

Distance is an important factor when considering application specific detectors. How close can you get the detector to the target radiation source. Worker safety and ALARA (as low as reasonably achievable) requirements, for example, emphasize the greatest distance from the source whenever possible. Perhaps you need a larger crystal...let's explore.

The Inverse Square Law shows an exponential increase in dose as you move closer to the source. It states:

Any point source which spreads its fluence equally in all directions without a limit to its range will obey the inverse square law.

This comes from strictly geometrical considerations. The intensity of the fluence at any given radius (r) is the source strength divided by the area of the sphere. Being strictly geometric in its origin, the inverse square law applies to diverse phenomena.

Point sources that obey the inverse square law include:

  • Gravitational force
  • Electric field
  • Light
  • Sound
  • Radiation

As one of the fields which obey the general inverse square law, a radiation point source can be characterized by the diagram below whether you are talking about Roentgens, rads, sieverts [1] or rems. All measures of exposure will drop off by the inverse square law. For example, if the radiation exposure is 100 mR/hr at 1 inch from a source, the exposure will be 0.01 mR/hr at 100 inches.

Notice that the diagram below only illustrates the principle of the inverse square law.  In practice the radiation from a point source is spherical (4π). Therefore, when this relationship is applied to a real application the detector must be in the shape of a right circular cylinder and the source should be positioned directly in front of the detector (on a centerline with the detector to avoid errors in geometry).  In addition, measurement errors in exposure (dose rate) must be minimized by positioning the radioactive source no closer than five times the diameter of the detector. This is because all photons impinging on the detector are not completely absorbed if the solid angle is too large.

Be aware that some detectors have shapes that don’t conform to the inverse square law when it comes to accurate dose rate measurements.  For example, a detector measuring 4"x4"x16" cannot accurately absorb all photons in a solid angle from a source at any position from this detector.  For these non-traditional detectors the factory calibration is usually set for compliance with NORM background.

[1] The sievert is the SI (International Unit) of exposure and is equal to 100rem.



Exponential Relationship for Dosage in mR/hr